ONLINE APPENDIX. Studying the Complexity of Rainfall within California via a Fractal Geometric Method

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1 A-1 ONLINE APPENDIX Studying the Complexity of Rainfall within California via a Fractal Geometric Method Carlos E. Puente 1, Mahesh L. Maskey 1 and Bellie Sivakumar 1, 2 Notation εε aaaa : Root mean square error in accumulated set εε mmmmmm : Maximum error in accumulated set νν jj : Nash-Sutcliffe efficiency on records at jj-day scale ηη jj : Nash-Sutcliffe efficiency on accumulated sets at jj-day scale ζ jj : Number of zeros present in the sets at jj-day scale ππ jj : Percent of zeros matched in the FM set at jj-day scale HH cc : Entropy of class distribution OOOO cc : Average orbit length by classes Table A1: Information about rainfall under study. Location Longitude Latitude Elevation Starting Period, Average (meters) date (years) (inches) Cherry Valley /10/ Merced /10/ Sacramento /10/ Shasta Dam /10/ Department of Land, Air and Water Resources, University of California, Davis Davis, CA 95616, USA cepuente@ucdavis,edu 2 School of Civil and Environmental Engineering, The University of New South Wales, Sydney, NSW 2052, Australia

2 A-2 Fig. A1: Examples of daily rainfall records in Merced for years 1905, 1950, 1980 and 2010 (black) and best FM representations (red), followed by their accumulated sets. Table A2: Performance of FM model for the best representations for Merced in Fig. A1 (εε aaaa, εε mmmmmm, ηη s, ππ s and νν s are in percent, ζ s are for data followed by FM fit in parenthesis). Statistics Period εε aaaa εε mmmmmm ηη ηη ηη νν νν νν ζ (323) 312(315) 301(310) 292(309) ζ 33 83(92) 89(93) 85(88) 80(86) ζ 77 28(30) 30(30) 28(28) 24(28) ππ ππ ππ

3 A-3 Fig. A2: Examples of daily rainfall records in Sacramento for years 1920, 1940, 1975 and 2005 (black) and best FM representations (red), followed by their accumulated sets. Table A3: Performance of FM model for the best representations for Sacramento in Fig. A2 (εε aaaa, εε mmmmmm, ηη s, ππ s and νν s are in percent, ζ s are for data followed by FM fit in parenthesis). Statistics Period εε aaaa εε mmmmmm ηη ηη ηη νν νν νν ζ (343) 308(314) 290(306) 292(256) ζ 33 94(104) 87(90) 78(87) 81(61) ζ 77 31(40) 29(31) 22(32) 22(19) ππ ππ ππ

4 A-4 Fig. A3: Examples of daily rainfall records in Shasta Dam for years 1945, 1975, 1995 and 2000 (black) and best FM representations (red), followed by their accumulated sets. Table A4: Performance of FM model for the best representations for Shasta Dam in Fig. A3 (εε aaaa, εε mmmmmm, ηη s, ππ s and νν s are in percent, ζ s are for data followed by FM fit in parenthesis). Statistics Period εε aaaa εε mmmmmm ηη ηη ηη νν νν νν ζ (253) 283(294) 249(283) 268(255) ζ 33 71(69) 81(82) 64(78) 71(67) ζ 77 23(22) 24(25) 22(27) 22(18) ππ ππ ππ

5 A-5 Fig. A4: Rainfall records in Merced for water years to (top-blue) and best FM representations (year by year-red) followed by their accumulated sets. The scales of the rain sets are in inches/day. Fig. A5: Rainfall records in Sacramento for water years to (top-blue) and best FM representations (year by year-red) followed by their accumulated sets. The scales of the rain sets are in inches/day.

6 A-6 Fig. A6: Rainfall records in Shasta Dam for water years to (top-blue) and best FM representations (year by year-red) followed by their accumulated sets. The scales of the rain sets are in inches/day. Fig. A7: Evolution of best FM parameters for Merced (blue) and averages over 5 years (red).

7 A-7 Fig. A8: Evolution of best FM parameters for Sacramento (blue) and averages over 5 years (red). Fig. A9: Evolution of best FM parameters for Shasta Dam (blue) and averages over 5 years (red). Table A5: Entropies of best FM parameters at all sites FM Parameters Site xx 1 xx 2 yy 1 yy 2 yy 3 dd 1 dd 2 pp φφ Cherry Valley Merced Sacramento Shasta Dam

8 A-8 Fig. A10: Evolution of three best FM parameters for Merced: first (red), second (blue) and third (green). Fig. A11: Evolution of three best FM parameters for Sacramento: first (red), second (blue) and third (green). Fig. A12: Evolution of three best FM parameters for Shasta Dam: first (red), second (blue) and third (green).

9 A-9 Table A6: Lower bounds on FM parameters for filtering purposes. FM Parameters Site xx 1 xx 2 yy 1 yy 2 yy 3 dd 1 dd 2 pp φφ Cherry Valley Merced Sacramento Shasta Dam Fig. A13: Evolution of filtered FM parameters for Merced (blue) and averages over 5 years (red). Fig. A14: Evolution of filtered FM parameters for Sacramento (blue) and averages over 5 years (red).

10 A-10 Fig. A15: Evolution of filter FM parameters for Shasta Dam (blue) and averages over 5 years (red). Table A7: Entropies of filtered FM parameters at all sites. FM Parameters Site xx 1 xx 2 yy 1 yy 2 yy 3 dd 1 dd 2 pp φφ Cherry Valley Merced Sacramento Shasta Dam Table A8: Complexity qualifiers of rainfall class evolutions. Classification based on Site Best FM Filtered FM Data deciles HH CC OOOO cc HH CC OOOO cc HH CC OOOO cc Cherry Valley Merced Sacramento Shasta Dam

11 A-11 Fig. A16: Pairwise site-comparison of filtered FM rainfall class evolutions and their scatter plots. The evolutions use blue for the set in the x-axis and red for the one in the y-axis. Sizes of circles are proportional to class repetitions. Fig. A17: Pairwise site-comparison of decile class evolutions and their scatter plots. The evolutions use blue for the set in the x-axis and red for the one in the y-axis. Sizes of circles are proportional to class repetitions.

12 A-12 Fig. A18: Complexity analysis of FM parameters for rainfall records in Cherry Valley, in order, parameter evolution, θθ(tt), parameter autocorrelation, ACF, and three phase-space diagrams, focusing on end-points of affine maps.

13 A-13 Fig. A18: (Cont.) Complexity analysis of FM parameters for rainfall records in Cherry Valley, in order, parameter evolution, θθ(tt), parameter autocorrelation, ACF, and three phase-space diagrams, focusing on vertical scalings, iterations proportion, vertical thresholds and total rainfall depth (in inches).

14 A-14 Fig. A19: Complexity analysis of FM parameters for rainfall records in Sacramento, in order, parameter evolution, θθ(tt), parameter autocorrelation, ACF, and three phase-space diagrams, focusing on end-points of affine maps.

15 A-15 Fig. A19: (Cont.) Complexity analysis of FM parameters for rainfall records in Sacramento, in order, parameter evolution, θθ(tt), parameter autocorrelation, ACF, and three phase-space diagrams, focusing on vertical scalings, iterations proportion, vertical thresholds and total rainfall depth (in inches).

16 A-16 Fig. A20: Complexity analysis of FM parameters for rainfall records in Shasta Dam, in order, parameter evolution, θθ(tt), parameter autocorrelation, ACF, and three phase-space diagrams, focusing on end-points of affine maps.

17 A-17 Fig. A20: (Cont.) Complexity analysis of FM parameters for rainfall records in Shasta Dam, in order, parameter evolution, θθ(tt), parameter autocorrelation, ACF, and three phase-space diagrams, focusing on vertical scalings, iterations proportion, vertical thresholds and total rainfall depth (in inches).